Mathematical analysis of motions of bodies in a viscous fluid

假设在粘性不可压缩流体中有一个刚体障碍物。对于物体与流体的相互作用分析，让我们考虑当物体以恒定角速度旋转时，物体外部的Navier-Stokes流动。我们首先证明了三维定常流的存在性和稳定性，并导出了在该定常流周围唯一整体解的大时间行为。对于时间周期流也得到了类似的结果。然后我们推导了空间无穷远定常流的衰减结构和渐近分布。给出了二维定常流的衰减结构，从而找到了Stokes佯谬的旋转解。

Suppose that there is an obstacle, which is a rigid body, in a viscous incompressible fluid. Toward analysis of body-fluid interaction, let us consider the Navier-Stokes flow in the exterior of the body when it is rotating with constant angular velocity. We first proved the existence and stability of the steady flow in 3D and derived the large time behavior of a unique global solution around that steady flow. The similar results were obtained for the time-periodic flow as well. We next deduced the decay structure and asymptotic profile of the steady flow at space infinity. The decay structure of the steady flow in 2D was also provided, so that we found the resolution of the Stokes paradox by the rotation of the body.